Đáp án:
\(A = 11 - 4\sqrt 7 \)
Giải thích các bước giải:
\(\begin{array}{l}
DK:x > 1\\
A = \dfrac{{\sqrt x - \sqrt {x - 1} - \sqrt x - \sqrt {x - 1} }}{{x - \left( {x - 1} \right)}} + \dfrac{{x\left( {\sqrt x - 1} \right)}}{{\sqrt x - 1}}\\
= \dfrac{{ - 2\sqrt {x - 1} }}{1} + x\\
= x - 2\sqrt {x - 1} \\
A = x - 1 - 2\sqrt {x - 1} .1 + 1\\
= {\left( {\sqrt {x - 1} - 1} \right)^2}\\
Thay:x = \dfrac{{53}}{{9 - 2\sqrt 7 }} = \dfrac{{53\left( {9 + 2\sqrt 7 } \right)}}{{81 - 28}}\\
= 9 + 2\sqrt 7 \\
\to A = {\left( {\sqrt {9 + 2\sqrt 7 - 1} - 1} \right)^2}\\
= {\left( {\sqrt {8 - 2\sqrt 7 } - 1} \right)^2}\\
= {\left( {\sqrt {7 - 2\sqrt 7 .1 + 1} - 1} \right)^2}\\
= {\left( {\sqrt {{{\left( {\sqrt 7 - 1} \right)}^2}} - 1} \right)^2}\\
= {\left( {\sqrt 7 - 1 - 1} \right)^2}\\
= {\left( {\sqrt 7 - 2} \right)^2}\\
= 7 - 4\sqrt 7 + 4 = 11 - 4\sqrt 7
\end{array}\)
